Solving Related Rates Problems - AP Calculus BC
Card 1 of 30
Identify the derivative of $\cot x$ with respect to $x$.
Identify the derivative of $\cot x$ with respect to $x$.
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$-\csc^2 x$. Negative cosecant squared function.
$-\csc^2 x$. Negative cosecant squared function.
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Find the derivative of $\csc x$ with respect to $x$.
Find the derivative of $\csc x$ with respect to $x$.
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$-\csc x \cot x$. Negative product of cosecant and cotangent.
$-\csc x \cot x$. Negative product of cosecant and cotangent.
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How do you express speed in terms of related rates?
How do you express speed in terms of related rates?
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$\frac{ds}{dt}$ where $s$ is displacement. Rate of change of position with time.
$\frac{ds}{dt}$ where $s$ is displacement. Rate of change of position with time.
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What is the derivative of $\ln x$ with respect to $x$?
What is the derivative of $\ln x$ with respect to $x$?
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$\frac{1}{x}$. Natural logarithm derivative is reciprocal.
$\frac{1}{x}$. Natural logarithm derivative is reciprocal.
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In related rates, how is acceleration expressed?
In related rates, how is acceleration expressed?
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$\frac{dv}{dt}$ where $v$ is velocity. Rate of change of velocity with time.
$\frac{dv}{dt}$ where $v$ is velocity. Rate of change of velocity with time.
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What is implicit differentiation?
What is implicit differentiation?
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Differentiation of equations not solved for one variable. Used when variables are mixed in equations.
Differentiation of equations not solved for one variable. Used when variables are mixed in equations.
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Determine the rate of change of volume of a cube with respect to its side length.
Determine the rate of change of volume of a cube with respect to its side length.
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$\frac{dV}{ds} = 3s^2$. Differentiating $V = s^3$ with respect to $s$.
$\frac{dV}{ds} = 3s^2$. Differentiating $V = s^3$ with respect to $s$.
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What is the implicit differentiation of $y^2 = x^3 + 3x$?
What is the implicit differentiation of $y^2 = x^3 + 3x$?
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$2y\frac{dy}{dx} = 3x^2 + 3$. Apply chain rule to both sides of equation.
$2y\frac{dy}{dx} = 3x^2 + 3$. Apply chain rule to both sides of equation.
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What is the chain rule for differentiation?
What is the chain rule for differentiation?
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$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. Differentiates composite functions step by step.
$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. Differentiates composite functions step by step.
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Which rule is used when differentiating quotients of functions?
Which rule is used when differentiating quotients of functions?
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Quotient Rule: $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$. Used for derivatives of function quotients.
Quotient Rule: $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$. Used for derivatives of function quotients.
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Identify the derivative of $\cot x$ with respect to $x$.
Identify the derivative of $\cot x$ with respect to $x$.
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$ - \csc^2 x $. Negative cosecant squared function.
$ - \csc^2 x $. Negative cosecant squared function.
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Identify the derivative of $\sin x$ with respect to $x$.
Identify the derivative of $\sin x$ with respect to $x$.
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$\cos x$. Sine and cosine are complementary derivatives.
$\cos x$. Sine and cosine are complementary derivatives.
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Which rule is used when differentiating products of functions?
Which rule is used when differentiating products of functions?
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Product Rule: $(uv)' = u'v + uv'$. Used for derivatives of function products.
Product Rule: $(uv)' = u'v + uv'$. Used for derivatives of function products.
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What is the derivative of $x^n$ with respect to $x$?
What is the derivative of $x^n$ with respect to $x$?
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$nx^{n-1}$. Power rule for polynomial differentiation.
$nx^{n-1}$. Power rule for polynomial differentiation.
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Identify the derivative of $\sec x$ with respect to $x$.
Identify the derivative of $\sec x$ with respect to $x$.
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$\sec x \tan x$. Product of secant and tangent functions.
$\sec x \tan x$. Product of secant and tangent functions.
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Identify the derivative of $\cos x$ with respect to $x$.
Identify the derivative of $\cos x$ with respect to $x$.
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$-\sin x$. Cosine derivative has a negative sign.
$-\sin x$. Cosine derivative has a negative sign.
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What is the derivative of $a^x$ with respect to $x$?
What is the derivative of $a^x$ with respect to $x$?
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$a^x \ln a$. Exponential rule with natural logarithm factor.
$a^x \ln a$. Exponential rule with natural logarithm factor.
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Find the rate of change of the hypotenuse in a right triangle.
Find the rate of change of the hypotenuse in a right triangle.
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Use $2c\frac{dc}{dt} = 2a\frac{da}{dt} + 2b\frac{db}{dt}$. Differentiate Pythagorean theorem with respect to time.
Use $2c\frac{dc}{dt} = 2a\frac{da}{dt} + 2b\frac{db}{dt}$. Differentiate Pythagorean theorem with respect to time.
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Find the derivative of $\csc x$ with respect to $x$.
Find the derivative of $\csc x$ with respect to $x$.
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$-\csc x \cot x$. Negative product of cosecant and cotangent.
$-\csc x \cot x$. Negative product of cosecant and cotangent.
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Find the rate of change of the area of a circle with respect to its radius.
Find the rate of change of the area of a circle with respect to its radius.
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$\frac{dA}{dr} = 2\pi r$. Differentiating $A = \pi r^2$ with respect to $r$.
$\frac{dA}{dr} = 2\pi r$. Differentiating $A = \pi r^2$ with respect to $r$.
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How do you find a related rate given a geometric relationship?
How do you find a related rate given a geometric relationship?
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Differentiate the relation with respect to time. Apply chain rule to connect rates through time.
Differentiate the relation with respect to time. Apply chain rule to connect rates through time.
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What is the derivative of $e^x$ with respect to $x$?
What is the derivative of $e^x$ with respect to $x$?
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$e^x$. Exponential function is its own derivative.
$e^x$. Exponential function is its own derivative.
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What is the related rates equation for a filling cone?
What is the related rates equation for a filling cone?
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Differentiate $V = \frac{1}{3} \pi r^2 h$ with respect to $t$. Apply chain rule to cone volume formula.
Differentiate $V = \frac{1}{3} \pi r^2 h$ with respect to $t$. Apply chain rule to cone volume formula.
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In a related rates problem, when should you solve for a variable?
In a related rates problem, when should you solve for a variable?
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After differentiating with respect to time. Substitute known values only after differentiating.
After differentiating with respect to time. Substitute known values only after differentiating.
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How do you express speed in terms of related rates?
How do you express speed in terms of related rates?
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$\frac{ds}{dt}$ where $s$ is displacement. Rate of change of position with time.
$\frac{ds}{dt}$ where $s$ is displacement. Rate of change of position with time.
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What is the chain rule for differentiation?
What is the chain rule for differentiation?
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$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. Differentiates composite functions step by step.
$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. Differentiates composite functions step by step.
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What is the implicit differentiation of $y^2 = x^3 + 3x$?
What is the implicit differentiation of $y^2 = x^3 + 3x$?
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$2y\frac{dy}{dx} = 3x^2 + 3$. Apply chain rule to both sides of equation.
$2y\frac{dy}{dx} = 3x^2 + 3$. Apply chain rule to both sides of equation.
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What is the derivative of $e^x$ with respect to $x$?
What is the derivative of $e^x$ with respect to $x$?
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$e^x$. Exponential function is its own derivative.
$e^x$. Exponential function is its own derivative.
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How do you find a related rate given a geometric relationship?
How do you find a related rate given a geometric relationship?
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Differentiate the relation with respect to time. Apply chain rule to connect rates through time.
Differentiate the relation with respect to time. Apply chain rule to connect rates through time.
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Find the rate of change of the area of a circle with respect to its radius.
Find the rate of change of the area of a circle with respect to its radius.
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$\frac{dA}{dr} = 2\pi r$. Differentiating $A = \pi r^2$ with respect to $r$.
$\frac{dA}{dr} = 2\pi r$. Differentiating $A = \pi r^2$ with respect to $r$.
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